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Directed Motion · If a function \(p(t)\) describes something’s position as a function of time, then \(v(t) = p'(t)\) describes its velocity, and \(a(t) = p''(t)\) describes its acceleration, and \(j(t) = p'''(t)\) describes its jerk, and \(s(t) = p^{(4)}(t)\) describes its snap (or jounce), and …
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Projectile Motion · If a projectile is launched from an initial height of \(y_0\) with initial upward velocity \(v_0\) and is subject to constant downward acceleration \(g,\) its height as a function of time can only be modelled by the formula \( y_0 + v_0t - \tfrac{1}{2}gt^2\,.\)
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Force and Momentum · The momentum \(p\) of an object is its velocity weighted by its mass. The force \(F\) experienced by an object is measured as its acceleration weighted by its mass. Force is the derivative of momentum with respect to time: \(F = \dot{p}\,.\)
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Work and Power · The physical quantity work \(W,\) which can also be thought of as “energy” and is measured in joules, is the accumulation of force with respect to displacement. The physical quantity power \(P,\) measured in watts (joules per second), is the rate at which energy is being transferred or converted. Power is the derivative of work (energy) with respect to time: \(P = \dot{W}.\)
- Charge and Current · An object’s charge \(Q\), measured in coulombs, is the physical property of matter that describes how it experiences force in an electromagnetic field. Current \(I\), measured in amps (coulombs per second), is a “flow” of charge through matter over time. I.e. current is the derivative of charge with respect to time: \(I = \dot{Q}\,.\)
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Economics & Finance ·
The phrase
marginal
implies a derivative is afoot. -
Population Dynamics ·
A population tends to grow at a rate proportional to its current size.
If \(P\) represents a population over time, then that phrase translates to the equation \(\dot{P} = kP\,.\) The function \(P\) that satisfies this equation has formula \(P(t) = \mathrm{e}^{kt}\,.\)